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Many computer programs contain algorithms that detail the specific instructions a computer should perform (in a specific order) to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Stored data are regarded as part of the internal state of the entity performing the algorithm.Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. that any procedure which could "naturally" be called effective, can in fact be realized by a (simple) machine. In practice, the state is stored in one or more data structures.So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming.This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means.

Subsequent formalizations were framed as attempts to define "effective calculability" those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 19, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation 1" of 1936, and Alan Turing's Turing machines of 1936–.

That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules.

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